Group of automorphisms for strongly quasi-invariant states

For a & lowast;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-automorphism group G on a von Neumann algebra, we study the G-quasi-invariant states and their properties. The G-quasi-invariance or G-strongly quasi-invariance is weaker than the G-invariance and has wide applications. We develop several properties for G-strongly quasi-invariant states. Many of them are the extensions of the already developed theories for G-invariant states. Among others, we consider the relationship between the group G and modular automorphism group, invariant subalgebras, ergodicity, modular theory, and abelian subalgebras. We provide with some examples to support the results.